Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#2044 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C24 | C24 |
| prime radical | C2×#534 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
1023 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 1023 conjugacy classes of subgroups — too many to draw. Only its 175 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 14 points:
(1,2);(4,5)(7,8)(9,11)(10,12)(13,14);(3,4)(5,6)(7,9)(8,11)(10,13)(12,14);(3,5,6,4)(11,14)52 conjugacy classes — view the full character table.
| maximal subgroup of | #137162, #137163, #137172, … |
| maximal normal subgroup of | #137162, #137163, #137172, … |
| maximal quotient of | C2×#10402, C2×#10404, C2×#10406, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16501);